The use of strongly minimal linearizations allows us to prove in Theorem  a much stronger result for the considered classes of structured polynomial matrices than those available in the literature for GLR-strong linearizations for the same structures (see, for instance, [1]}, [2]}, [3]}, [4]}, [5]}, [6]}). We emphasize that Theorem  provides a simple recipe for constructing a structure preserving strongly minimal linearization for any polynomial matrix with the considered structures. In contrast, such generality is not possible by using GLR-strong linearizations. For instance, it is shown in [5]}, [6]} that there exist para-symmetric and para-skew-symmetric polynomial matrices of even degree for which there do not exist any GLR-strong linearization with the same structure. Moreover, it is proved in [2]} that structure preserving GLR-strong linearizations of size \(dm \times dm\)  do not exist for \(m \times m\)  polynomial matrices \(P(\lambda )\)  of even degree \(d\)  possessing any of the self-conjugate structures studied in this paper whenever \(P(\lambda )\)  has an odd number of left minimal indices and an odd number of right minimal indices. Note that \(dm \times dm\)  is precisely the size of all explicitly and easily constructible families of GLR-strong linearizations available in the literature for \(m\times m\)  matrix polynomials of degree \(d\) , as, for instance, “vector space linearizations” [10]}, [4]}, [12]}, [5]}, “Fiedler-like linearizations” [14]}, [1]}, [16]} and “block Kronecker linearizations” [17]}, [3]}. Thus, the result in [2]} proves that there exist self-conjugate structured polynomial matrices that cannot be linearized in these families in a structure preserving way. In fact, some of these families of GLR-strong linearizations present other drawbacks as, for instance, the ones considered in [4]}, [5]} are not valid for any singular polynomial matrix [22]} or require certain nonsingularity conditions on the coefficients of the polynomial matrix or on the location of its eigenvalues.
